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Understanding Reasoning in LLMs through Strategic Information Allocation under Uncertainty

Jeonghye Kim, Xufang Luo, Minbeom Kim, Sangmook Lee, Dongsheng Li, Yuqing Yang

Abstract

LLMs often exhibit Aha moments during reasoning, such as apparent self-correction following tokens like "Wait," yet their underlying mechanisms remain unclear. We introduce an information-theoretic framework that decomposes reasoning into procedural information and epistemic verbalization - the explicit externalization of uncertainty that supports downstream control actions. We show that purely procedural reasoning can become informationally stagnant, whereas epistemic verbalization enables continued information acquisition and is critical for achieving information sufficiency. Empirical results demonstrate that strong reasoning performance is driven by uncertainty externalization rather than specific surface tokens. Our framework unifies prior findings on Aha moments and post-training experiments, and offers insights for future reasoning model design.

Understanding Reasoning in LLMs through Strategic Information Allocation under Uncertainty

Abstract

LLMs often exhibit Aha moments during reasoning, such as apparent self-correction following tokens like "Wait," yet their underlying mechanisms remain unclear. We introduce an information-theoretic framework that decomposes reasoning into procedural information and epistemic verbalization - the explicit externalization of uncertainty that supports downstream control actions. We show that purely procedural reasoning can become informationally stagnant, whereas epistemic verbalization enables continued information acquisition and is critical for achieving information sufficiency. Empirical results demonstrate that strong reasoning performance is driven by uncertainty externalization rather than specific surface tokens. Our framework unifies prior findings on Aha moments and post-training experiments, and offers insights for future reasoning model design.
Paper Structure (57 sections, 2 theorems, 42 equations, 12 figures, 5 tables)

This paper contains 57 sections, 2 theorems, 42 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Understanding Aha moments.
  4. Theoretical Understanding of Reasoning.
  5. Theoretical Unification: Reasoning as Strategic Information Allocation
  6. Reasoning as Self-Conditioning
  7. Limits of Procedural Information
  8. Externalized Uncertainty as Information
  9. Limits of Token-Level Uncertainty
  10. Epistemic Verbalization
  11. Epistemic Verbalization for Continued Information Acquisition
  12. Self-Correction as a Control Action
  13. Informational Role of Epistemic Verbalization
  14. Epistemic Verbalization and MI Peak
  15. Relationship Between Uncertainty and Epistemic Verbalization
  16. ...and 42 more sections

Key Result

Lemma 3.5

Let $\hat{Y}_T = g_T(\tilde{S}_T)$ be any estimator of the target variable $Y$, where $\tilde{S}_T$ denotes the random variable corresponding to the augmented reasoning state at step $T$, and define the error probability $P_e(T) := \Pr[\hat{Y}_T \neq Y].$ Assume $|\mathcal{Y}| < \infty$. If $P_e(T)

Figures (12)

  • Figure 1: We identify three common modes of reasoning collapse in procedural reasoning: (1) Recursive step expansion, where the solver gets stuck and resorts to brute-force substitutions or repetitive steps; (2) Problem injection, where the solver implicitly shifts to solving a different problem; and (3) Degenerate loops, where the solver repeatedly generates the same words, tokens, or structures without making progress. Concrete examples are provided in Appendix \ref{['appendix:qualitative']}.
  • Figure 2: Token-level entropy over reasoning steps for Qwen2.5-Math-7B and Qwen3-14B-Base on AIME24 decreases similarly in both correct and incorrect solutions, suggesting that entropy alone does not reliably reflect uncertainty toward the correct answer.
  • Figure 3: On AIME24 #7, both models initially failed, but only Qwen3-8B-Base-SFT sustained information gain via epistemic verbalization and self-corrected to the correct answer.
  • Figure 4: Token-level analysis of MI shows that high MI corresponds to evaluative behaviors rather than the tokens themselves.
  • Figure 5: Comparison of epistemic token prevention in DeepSeek-Distill-32B/14B and induction in Qwen3-14B-Base.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Definition 3.1: Reasoning Objective
  • Definition 3.2: Information Gain of a Reasoning Step
  • Definition 3.4: Augmented Reasoning State
  • Lemma 3.5: Information Sufficiency as a Necessary Condition
  • Proposition 3.6: Sporadic Epistemic Verbalization Enables Continued Information Acquisition